Limit distributions of norms of vectors of positive i. i. d. random variables
type: Item
instance of: scholarly article
title: Limit distributions of norms of vectors of positive i. i. d. random variables
zbMATH Open document ID: 1014.60015
author: M. Schlather
published in: The Annals of Probability
publication date: 2003-05-06T00:00:00Z
review text: Let \((X_n)_{n\in\mathbb N}\) be an i.i.d.~sequence of positive random variables, \(c\) a positive constant and \((p(n))_{n\in\mathbb N}\) a sequence in \((0,\infty]\). The present paper is concerned with convergence of the suitably centered and normalized \(c p(n)\)-norm of the vector \((X_1, \ldots, X_n)\) to a limit distribution \(F_c\). The limit distributions \(F_c\) are determined in the case that \(X_1\) is in the maximum domain of attraction of the Fréchet distribution, where \(p(n)=1\), and in the case that \(X_1\) is in the maximum domain of attraction of the Weibull distribution, where \((p(n))_{n\in\mathbb N}\) is a sequence depending on the upper tail behaviour of \(X_1\), but not on \(c\). Centering and normalization constants are given explicitly. It is shown that in both cases, \(F_c\) converges in distribution to the standard normal distribution as \(c \to 0+\), and to the Fréchet or Weibull distribution, respectively, as \(c \to \infty\). For sequences in the maximum domain of attraction of the Gumbel distribution, the example of exponential random variables is considered, and conjectures for the general case are given.
Mathematics Subject Classification ID: 60F05
Mathematics Subject Classification ID: 60G50
Mathematics Subject Classification ID: 60G70
Mathematics Subject Classification ID: 60E07
zbMATH DE Number: 1906003
zbMATH Keywords: central limit theorem
zbMATH Keywords: extreme value theory
zbMATH Keywords: independent identically distributed positive random variables
zbMATH Keywords: \(l_p\)-norm
zbMATH Keywords: limit theorems
zbMATH Keywords: normal distribution
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aop/1008956695
OpenAlex ID: W1895320873
cites work: Q3785143
cites work: Q4718576
cites work: Q4836494
cites work: Q4726487
cites work: Q3862781
cites work: Q4343010
cites work: Q5624436
cites work: THE WORK OF A. N. KOLMOGOROV IN THE THEORY OF PROBABILITY
cites work: Q4515165
cites work: On the domain of attraction of an operator between supremum and sum
cites work: Some extensions of the LIL via self-normalizations
cites work: Asymptotic behavior of self-normalized trimmed sums: Nonnormal limits
cites work: On the asymptotic joint distribution of the sum and maximum of stationary normal random variables
cites work: The domain of attraction of the \(\alpha\)-sun operator for type II and type III distributions
cites work: Large deviations and law of the iterated logarithm for partial sums normalized by the largest absolute observation
cites work: A note on the asymptotic independence of the sum and maximum of strongly mixing stationary random variables
cites work: Foundations of Modern Probability
cites work: Extremes and related properties of random sequences and processes
cites work: Limit distributions of self-normalized sums
cites work: Q3771297
cites work: Q4301585
cites work: Q4937701
cites work: Self-normalized large deviations
DOI: 10.1214/AOP/1008956695
description: scientific article; zbMATH DE number 1906003